. . Affine space fibrations . . . . . M. Miyanishi jointly with R.V. Gurjar and K. Masuda RCMS, Kwansei Gakuin University Warsaw, May 28, 2018 . . . . . . M. Miyanishi jointly with R.V. Gurjar and K. Masuda Affine space fibrations
. Contents of the talk 0. Outline 1. Singular fibers of A 1 -fibrations on algebraic surfaces 2. Singular fibers of P 1 -fibrations on algebraic surfaces 3. Singular fibers of A 1 - and P 1 -fibrations on algebraic threefolds 4. Equivariant Abhyankar-Sathaye conjecture in dimension three 5. Forms of A n and A n × A 1 ∗ with unipotent group actions k = k : the ground field of characteristic zero. A dominant morphism f : X → Y of algebraic varieties is a fibration equivalently if the generic fiber X η is geometrically integral, general fibers are integral, k ( X ) ⊃ k ( Y ) is a regular extension, k ( Y ) is algebraically closed in k(X). . . . . . . M. Miyanishi jointly with R.V. Gurjar and K. Masuda Affine space fibrations
. § 0. Outline Let F be an algebraic variety. We say that f is an F -fibration if X y is isomorphic to F over k for general (closed) points y of Y . The F -fibration is locally trivial if f − 1 ( U ) � U U × F for an open set U � ∅ of Y . f is locally isotrivial if f − 1 ( U ) × U U ′ � U ′ U ′ × F for an etale covering U ′ → U . If f is locally trivial open set U and a finite ´ (or locally isotrivial) for an open neighborhood U y (or a ´ etale finite covering U ′ y of U y ) of each closed point y then X is an F -bundle over Y (or an ´ etale F -bundle over Y ). Let X η := X × Y Spec k ( Y ) be the generic fiber of f . Then X η � k ( Y ) F (or X η � K F for an algebraic extension K / k ( Y ) ) if f is locally trivial (or locally isotrivial). A fundamental question is if a fibration f : X → Y is locally trivial or locally isotrivial provided X y � F for gneral points y ∈ Y . . . . . . . M. Miyanishi jointly with R.V. Gurjar and K. Masuda Affine space fibrations
The generic isotriviality of an F -fibration for an affine variety F follows from . Gereric Equivalence Theorem of Kraft-Russell . . . Let k be an algebraically closed field of infinite transcendence degree over the prime field. Let p : S → Y and q : T → Y be two affine morphisms where S , T and Y are k -varieties. Assume that for all y ∈ Y the two (schematic) fibers S y := p − 1 ( y ) and T y := q − 1 ( y ) are isomorphic. Then there is a dominant morphism of finite degree φ : U → Y and an isomorphism S × Y U � U T × Y U . . . . . . If f : X → Y is an F -fibration, the generic fiber X η is a k ( U ′ ) / k ( Y ) -form of F . The triviality of the form X η is the generic local triviality of f . If F = A n , an F -fibration is an affine space fibration. The local triviality is a Dolgachev-Weisfeiler problem. If n = 1 , 2 the answer is affirmative. . . . . . . M. Miyanishi jointly with R.V. Gurjar and K. Masuda Affine space fibrations
An F -fibration f : X → Y has singular fiber which is, by definition, a closed fiber X y which is not isomorphic to F . There are four possibilities for which the fiber X y is not isomorphic to F . (1) The fiber X y is integral, but not isomorphic to F . (2) The fiber X y is not integral. Hence either X y has two or more irreducible components (reducible fiber), or X y is irreducible but non-reduced (non-reduced fiber). (3) Each irreducible component Z i of X y has right dimension dim F , but has multiplicity length O X y ,ξ i > 1 which is a multiple of some integer d ≥ 1 (multiple fiber), where ξ i is the generic point of Z i . (4) Some irreducible component Z i has dimension bigger than dim F . . . . . . . M. Miyanishi jointly with R.V. Gurjar and K. Masuda Affine space fibrations
. § 1. Singular fibers of A 1 -fibrations on algebraic surfaces Let f : X → C be an A 1 -fibration on an algebraic surface X and let F be a fiber of f . To study F , the following reduction is possible. (1) The curve C is smooth with C replaced by the normalization C and X by X × C � � C . (2) f is an affine morphism with C replaced by an affine open nbd U of P := f ( F ) and X by f − 1 ( U ) . (3) f is the quotient morphism by a G a -action on X . Then our problem is : . Problem 1.1 . . . Is every fiber F := f − 1 ( P ) for P ∈ C a disjoint union of the affine lines? . . . . . . . . . . . M. Miyanishi jointly with R.V. Gurjar and K. Masuda Affine space fibrations
Our present knowledge is: . Theorem 1.2 . . . (1) F is a disjoint union of the irreducible components, each of which is an affine rational curve with one place at infinity. (2) If an irreducible component Z i of F is reduced in F , then Z i � A 1 . (3) If X is normal, F is a disjoint union of the affine lines. Every singular point on X is a cyclic quotient singularity. . . . . . . Theorem 1.3 . . . Let f : X → C be a dominant morphism from an affine surface X to an affine curve C . Assume that, for every closed point P ∈ C , the fiber f − 1 ( P ) is a disjoint union of the affine lines. Then there exists a nontrivial G a -action on X such that the morphism f is factored by the quotient morphism q : X → X / G a as q g f : X − → X / G a − → C , where g is a quasi-finite morphism. . . . . . . . . . . . M. Miyanishi jointly with R.V. Gurjar and K. Masuda Affine space fibrations
. § 2. Singular fibers of P 1 -fibrations on algebraic surfaces Let f : X → C be a P 1 -fibration from an algebraic surface to an algebraic curve. We may assume that C is normal. The generic local triviality follows from Tsen’s Theorem. . Lemma 2.1 . . . Suppose that X is normal. Then we have: (1) X has only rational singularities, whose resolution graph is a tree of smooth rational curves and is a part of a degenerate fiber of a P 1 -fibration on a smooth surface. (2) Every fiber F of f is a union of smooth rational curves, and its intersection dual graph is a tree in the sense that the dual graph of the inverse image of F in a minimal resolution of singularity of S is a tree. (3) H 1 ( F ; Z ) = 0 . . . . . . . . . . . . M. Miyanishi jointly with R.V. Gurjar and K. Masuda Affine space fibrations
. Lemma 2.2 . . . Let f : X → C be a P 1 -fibration over a normal curve C . Let F be a fiber of f . Then we have. (1) The singular locus of X is contained in the union of finitely many fibers of f . (2) F is a connected union of rational irreducible components. (3) π 1 ( F ) is a cyclic group. If X is normal, F is simply-connected. . . . . . . Lemma 2.3 . . . Let f : X → C be a projective morphism which is a P 1 -fibration over a smooth curve C and let F be a fiber of f . Then F is simply connected. In particular, ecery irreducible component is homeomorphic to P 1 . . . . . . . . . . . . M. Miyanishi jointly with R.V. Gurjar and K. Masuda Affine space fibrations
§ 3. Singular fibers of A 1 - and P 1 -fibrations on algebraic threefolds . One of the natural looking but hard to prove results is: . Theorem 3.1 . . . Let X be a smooth affine threefold with a G a -action and let q : X → Y be the quotient morphism. Assume that Y is smooth. Let F = q − 1 ( P ) be a fiber and write F = Γ + ∆ , where Γ (resp. ∆ ) is pure 1 - (resp. 2 -) dimensional. Then we have: (1) Γ is a disjoint union of A 1 s and Γ ∩ ∆ = ∅ . (2) Let S be a component of ∆ . Let L be a general hyperplane section of Y through the point P and let T be the closure in X of q − 1 ( L \ { P } ) . Assume that T ∩ S � ∅ . Then S has an A 1 -fibration parallel with f . . . . . . . . . . . . M. Miyanishi jointly with R.V. Gurjar and K. Masuda Affine space fibrations
Turning to a P 1 -fibration f : X → Y with dim X = n and dim Y = n − 1 , we first note that f is not necessarily generically locally trivial if n ≥ 3 . . Lemma 3.2 . . . Suppose that X and Y are smooth. Then we have: (1) Let S be the closure of the set of points Q ∈ Y such that either the F Q := X × Y Spec k ( Q ) has an irreducible component of dimension > 1 or every irreducible component F i of F Q has multiplicity > 1 , i.e., length O F Q , F i > 1 . Then codim Y S > 1 . (2) Let n = 3 . Then every fiber F Q is simply-connected. (3) Let n = 3 and write F Q = Γ + ∆ as in the case of A 1 -fibration. Then H 1 (∆; Z ) = H 1 (Γ; Z ) = 0 . Each component of Γ is a rational curve, and each component of ∆ is a rational surface or a rationally ruled surface. . . . . . . . . . . . M. Miyanishi jointly with R.V. Gurjar and K. Masuda Affine space fibrations
If n = 2 we have: . Lemma 3.3 . . . Let f : X → C be a P 1 -fibration over a smooth curve C and let F be a singular fiber of f . Then F is simply connected. In particular, ecery irreducible component is homeomorphic to P 1 . . . . . . In general, we have: . Theorem 3.4 . . . Let f : X → Y be an equi-dimensional P 1 -fibration over a smooth variety Y and let F be a closed fiber of f . Then F is simply-connected. . . . . . . . . . . . M. Miyanishi jointly with R.V. Gurjar and K. Masuda Affine space fibrations
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